Observation of beta-induced Alfvén eigenmodes in the DIII-D tokamak

Energetic ions can drive Alfven gap modes unstable, causing large losses of fast ions. Toroidicity-induced Alfven eigenmodes (TAE) were expected to disappear into the shear Alfven continuum and become stable as the plasma beta increased. Although TAE modes may disappear, another dangerous instability with similar properties but approximately half the TAE frequency appears in a spectral gap that is created by finite beta eff'ects. The measured frequency of the new mode agrees with the theoretical frequency of beta-induced Alfven eigenmodes.

The center of this "TAE gap" occurs at a frequency OAF = Vg/4tE'qR, where vz =B/ J4trn;m; is the Alfven speed, q is the safety factor, R is the tokamak major radius, B is the magnetic field, and n; and m; are the ion density and mass [5].
(An example of the gap structure appears in Fig. 4.) Analysis of the TAE mode suggested that its frequency would decrease into the Alfven continuum (where it would be heavily damped) as the plasma beta approached the stability limit for ideal ballooning modes [6].But re- cent calculations found a new gap underneath the Alfven continuum caused by the compressional response of the plasma to shear Alfven waves in the presence of finite pressure and curvature [7].The energy associated with this compression produces a frequency shift that raises the Alfven continuum, thus opening a low-frequency gap.Global modes in this gap with the dominant polarization of shear Alfven waves were discovered numerically [8]; we call these modes beta-induced Alfven eigenmodes (BAE).In this Letter, experimental evidence of destabil- ization of BAE modes by energetic beam ions is reported for the first time.
The experiments are performed in the DIII-D tokamak (R= 1.8 m, a =0.65 m) in relatively pure (Z,tr ~2) deu- terium plasmas.Near tangential (tangency radius R"" =1.10 m), -75 keV deuterium neutrals are injected in the direction of the plasma current; in some plasmas, near-perpendicular beams (R&,"=0.74 m) are also inject- ed.The normalized beta, Ptv -= PtaB, /Iz, is usually near the nominal limit of Ptv=3.5, although larger values can be obtained with current ramping.
(Here P, is the toroidal beta in percent, a is the minor radius in meters, B, is the toroidal field in tesla, and I~i s the plasma current in MA. ) Most of the discharges discussed here have an elongation of tr=1.6 and use the inner wall as a limiter, although divertor discharges exhibit similar behavior.The principal diagnostic for the study is an exten- sive array of magnetic probes mounted inside the vacuum vessel.Toroidal mode numbers n are obtained from the best fit to the phase differences of a toroidal array of eight probes.Electron temperature and density profiles are measured by Thomson scattering, the ion temperature and toroidal rotation profile is determined from charge exchange recombination spectroscopy of Heal, and the q profile is obtained from the magnetic configuration, the soft x-ray inversion radii, and a single-point ineasurement of q using motional Stark effect polarimetry.
Figure 1 shows the evolution of the frequency of beam-driven instabilities as the beam power is increased in increments of -2.5 M%'.The frequency is corrected for the Doppler shift, as discussed below.With only one source (2.5 MW), no beam-driven instabilities are ob- served.After a second source is added at 1.7 s, TAE modes appear.At some times, two sets of peaks appear in the power spectrum simultaneously.
Theoretical calculations indicate that several TAE modes with different frequencies can occupy the TAE gap, and the measured frequencies of the multiple modes agree with the calcula- tions [8].During this two-beam phase, the normalized beta is ptv=2. 8.After the third beam is injected at 1.9 s, Ptv increases to 3.3 and the frequency of the beam-driven modes drops well below the nominal TAE frequency.As we will show, these instabilities are probably BAE modes.With the addition of the fourth beam at 2. 1 s, the plasma reaches the nominal beta limit (Ptv =3.5) for these condi- 0-  [9].) The mode frequency tends to decrease with increasing Pjv, as shown in Fig. 2. In this figure, the Doppler- corrected frequency is normalized to fTAF.The data are from sixty discharge conditions that span the parameter range I~= 0.4-1.0MA, 8& =0.7-1.4T, n, = (1.1-5.5) x 10' cm, Pb =2.5-16 MW, x'=1.1-2. 2,and P& =1.5%-6.7%.For modest values of normalized beta (P~(25), the measured frequency agrees with fTAF within -25%.Three known sources of error contribute to the scatter in f/fTAF.First, the uncertainty in the Doppler-shift correction is -10%.Second, the experi- mental value of q need not equal the value of 1.5 assumed in fTAE.Third, the actual tnode frequency does not necessarily lie in the center of the gap, as assumed by Eq.
(1).For a toroidal field scan where the changes in plas- ma profiles are minimized, the scatter in f/fT~E is re- duced to 12%.
As the normalized beta increases, the ratio of f/fTAE drops and the scatter becomes larger (Fig. 2).At the nominal beta limit of P~=3.5, the frequency is usually less than half the expected TAE frequency, although oc- casionally larger values are observed (as in Fig. 1 after 2. 16 s).The data with P~=5.2 in Fig. 2 are from a sin- gle discharge with a strong negative current ramp, so it is not certain if such a low value of f/fTAE is always ob- served for these conditions.The decrease of f/fTAE with P~i s one of the strongest dependences observed in our data set (the correlation with the poloidal beta P~i s simi- the nominal TAE frequency [Eq.(I)], and the beam power in shot 71524.The Doppler shift is obtained from the spacing of peaks with different mode numbers, as in Fig. 3(b); the uncer- tainty in frequency is typically 5 kHz.The gradual decrease in fTAa is caused by an increase in electron density during beam injection.[Here, and in Figs. 2 and 5, the nominal values q =1. 5, R =1.7 m, n; =n"and 8 =8, are used for fr~E in Eq.
The Doppler-shift correction for TAE modes has been studied extensively and will be reported elsewhere.In brief, the correction bco can be obtained either from mea- surements of the toroidal rotation profile or from fits to the toroidal mode number spectrum.The first method as- sumes that helium impurities rotate with the same veloci- ty as the bulk deuterium plasma and that the poloidal ro- tation velocity Uz is much smaller than the toroidal rotation v& so that 6ro=nv&/R The sec.ond method assumes that TAE modes with diA'erent toroidal mode numbers rotate with the same speed v& and have identical frequen- cies in the plasma frame.The two methods agree well (to within -10%) and with theoretical calculations of the TAE frequency [3].Since the structure of BAE modes resembles the structure of TAE modes (see below), these techniques are valid for BAE modes as well.(The en- velope of the curves is similar for higher values of n [7].) beams causes the plasma to rotate toroidally at -11 kHz and introduces Doppler shifts in the spectrum.Three sets of peaks are observed.The first set of low-frequency peaks (labeled "Mirnov") are the usual set of peaks asso- ciated with MHD activity that is nearly stationary in the plasma frame.A plot of the toroidal mode number n of these peaks versus the observed frequency falls on a straight line; this line intercepts the frequency axis near the origin [Fig.3(b)].In this discharge, the TAE modes appear above 100 kHz.The Doppler-corrected frequency inferred from the toroidal mode number spectrum [the frequency-axis intercept in Fig. 3(b)] is 58 kHz.The peaks between 50 and 100 kHz are the new feature asso- ciated with BAE modes.The BAE modes have toroidal mode numbers that are similar to TAE modes (typically n =2-8) and the mode number spectrum intercepts the frequency axis at 22 kHz.
Comparison of the measured frequencies with the theoretical spectrum of Alfven modes is consistent with identification of the low-frequency peaks as BAE modes.
Figure 4 shows the Alfven continuum spectrum calculated by the coNT code [7] for n =3 modes.Finite beta creates an additional gap beneath the lowest continuum branch.It also complicates the spectrum by coupling the shear Alfven waves to sound waves.This coupling is re- sponsible for the sharp jumps in the continuum curves in Fig. 4; nevertheless, the general envelope of the Alfven continuum is evident.Also shown as horizontal lines in Fig. 4 are the frequencies calculated by the ideal MHD stability code GATO for n =3 BAE and TAE modes.The calculated BAE modes have radially extended eigenfunc- tions which peak in the plasma interior.These modes have the dominant polarization of Alfven waves (g) gii, where g is the displacement vector) [8].
The Doppler-corrected experimental mode frequency is shown as a function of minor radius in Fig. 4 for several beam-driven modes.Here the plasma rotation speed profile v&(r) is used to calculate the mode frequency in the frame rotating toroidally with the plasma at radius r.
This Doppler-corrected frequency should equal the theoretical frequency at the radial location of the mode.
The Doppler-corrected frequencies agree with the calculated BAE frequency near the q = 1 surface, where the BAE frequency calculated by GATO lies in the beta- induced spectral gap and where the calculated BAE eigenfunction has its maximum amplitude.
Although the experimental curves also pass through the toroidicity-induced gap between the q =1.5 surface and the plasma edge, they probably do not represent TAE modes excited in the outer part of the discharge.GATO  calculations indicate that TAE modes can exist in this edge region, but the eigenfunctions peak near the q =2.5 surface and do not extend into the plasma interior.Since the calculated fast-ion driving rate is very small outside the q =2 surface [3], these edge modes are probably stable.(At lower values of Piv, the calculated TAE modes that agree with experiment have eigenfunctions that peak in the interior [8].)Soft x-ray measurements confirm that the BAE modes are excited in the center of the plasma.For these experiments, the two soft x-ray ar- rays measure inward displacements and vertical displacements.Analysis of the data indicates displacements of G(0.1-1) mm, with similar values observed for the high- frequency (TAE) modes and the low-frequency (BAE) modes.In both cases, the profile is globally extended and the largest displacements occur for q & 1.5.
The frequency of BAE modes increases linearly with toroidal field, as expected for Alfven waves.Figure 5 shows the Doppler-corrected frequency as a function of Alfven speed for a scan of toroidal field in discharges with similar shapes and densities.The data during heat- ing with two sources (Pb=5 MW) agree to within 20%%uv with the expected scaling for TAE modes [3].During the four-beam phase (Pb=10 MW), the frequency still in- creases with increasing Alfven speed, but the slope is as large.The frequency of BAE modes calculated by GATO also increases linearly with vz and is consistent with the experimental observations.In contrast, the data do not scale with co~"with JT;, or with JT"as would be expected for drift waves, acoustic modes, or semicol- V"(10 m/s) FIG. 5. Doppler-corrected frequency versus the Alfven speed during a scan of the toroidal field from 0.6 to 1.4 T in   discharges like shot 71524.The error bars indicate the stan- dard deviation of the frequency for several times near 1.85 s (triangles) or 2. 15 s (solid).At IO MW, Ptv varies from 3.0 to 3.5.Also shown is the nominal TAE frequency (solid line) and the frequencies of BAE modes calculated by GATo (circles).
The ion diamagnetic frequency (x) is evaluated at the q=l surface using spline fits to the measured T;, n"and Z,q profiles and the approximation ke=nq/r, where n is the toroidal mode number of the largest amplitude mode and r is the minor radius.lisional Alfven modes [10].The frequency also does not scale with the ion diamagnetic frequency to+~; (Fig. 5).
Nominally, the frequency of kinetic ballooning modes scales with co~~; [11],although recent work suggests the predicted mode frequency falls to to+~;/2 near the beta limit [12].Although the data do fall within this range, the linear scaling with U~suggests an Alfven wave.
23 power spectra with multiple sets of peaks have been observed (cf.Fig. 3).For the cases where the frequency of the strongest mode is comparable to fyAE, the frequen- cies of the two modes differ by (12 6)%, indicating that both modes lie in the TAE gap.In contrast, when the frequency of the strongest mode is ( -, ' f~p, E, the fre- quencies differ by (44 ~15)%.For these cases, the stronger mode lies in the BAE gap, while the weaker mode resides in the TAE gap.
Beam-driven instabilities usually "saturate" in a relax- ation cycle about the point of marginal stability; the satu- ration occurs when the fast ions that drive the instability are expelled from the plasma by the instability [13].
Analysis of the nonlinear cycle yields information about the damping rate of the instability and about the mecha- nism of fast-ion loss [13].When the frequency of the un- stable mode drops from the TAE gap to the BAE gap as in Fig. 1 at 1.91 s, the nonlinear cycle does not suddenly change but gradually evolves.This suggests that the damping rate and particle losses do not diAer greatly for the two instabilities.
The drop in neutron emission caused by beam-ion loss is comparable for TAE and BAE modes with similar mode amplitudes.These observations are consistent with the hypothesis that both instabilities are Alfven gap modes with spatially extended eigenfunc- tions.
In summary, a beam-driven instability with a frequen- cy less than half the TAE frequency is observed as the plasma approaches the beta limit.The frequency of the instability lies below the gap in the Alfven continuum occupied by TAE modes, in a new gap that is created by finite beta effects.The following observations suggest that the instability is a BAE mode.(1) The mode fre- quency scales linearly with the Alfven speed.(2) The mode is only observed at large Ptv.(3) The mode fre- quency agrees well with the calculated frequency of BAE modes and lies in the BAE gap.(4) The mode structure, nonlinear cycle, and fast-ion losses are similar to TAE modes.
The observation of these modes dashes hopes that Alfven instabilities can be avoided by operating near the beta limit.Experimentally, the BAE mode seems just as deleterious as the TAE mode.Future theoretical and ex- perimental work should concentrate on understanding the stability properties of this new instability.Control of BAE modes in a reactor may require operation in a re- gime that minimizes the alpha-particle pressure (low T, and high n, ) or modification of the Alfven gap structure to increase mode damping [3].
The contributions of the DIII-D team are gratefully acknowledged.K. H. Burrell, E. Carolipio, H. H. Duong, and E. Wilfrid assisted with the data analysis.This work was supported by Subcontract No. SC-L134501 of the U.S. Department of Energy Contract No. DE-AC03- 89ER51114.

FIG. 2 .
FIG.2.Doppler-corrected frequency divided by frAE as a function of PN for many discharges.The error bars represent the standard deviation of the data.

Figure 3 (FIG. 3 .FIG. 4 .
Figure3(a) shows the cross-power spectrum of two toroidally separated magnetic probes at 2. 177 s in the discharge of Fig.1.The momentum input from the